Kerry K. Kuehn

Galileo, Harmony and Piano Tuning

In his Dialogues Concerning Two New Sciences (1638), Galileo raises the question: why it is that certain combinations of notes sound good together, while others do not?

For example, two notes separated by an octave, or three notes forming a major chord, will sound pleasant when struck simultaneously. On the other hand, if one were to randomly pick out two keys on a piano and strike them simultaneously, then they would typically sound dissonant. Why is this?

Perhaps most interestingly, it seems that the identification of "pleasing" or "unpleasing" combinations of notes is not a subjective process, in the sense that people throughout history and from vastly different cultures come to similar conclusions on this topic. Why might this be?

The key to unlocking this mystery, according to Galileo, is in recognizing music as one of the four liberal arts that make up the quadrivium. The quadrivium, according to ancient and medieval custom, consisted of arithmetic, geometry, music and astronomy. Despite the apparent difference in these subjects, what they share is the unifying theme of numbers and their ratios.

For example, arithmetic is the elementary study of how to manipulate numbers and their ratios. Geometry is then the application of these concepts to shapes and figures, such as triangles, squares and circles. Astronomy is the study of ratios as they relate to the motion of the planets, sun, moon and stars. And music, finally, is the study of ratios as they are perceived by the ear. All of the subjects of the quadrivium involve the notion of harmony.

So when Galileo approaches the problem of musical harmony, he begins by arguing that each note is associated with a particular number denoting its frequency of vibration. Now if the vibration frequencies of two (or three or more) notes can be expressed as a ratio of small integers, then the combination of notes is pleasant. For example, two notes separated by an octave have a frequency ratio of 2:1. Two notes separated by fifth have a frequency ratio of 3:2. And the three notes making up a C chord have a frequency ratio of 4:5:6.

According to Galileo, the smaller the numbers in these ratios, the more consonant the sound. So two strings separated by an octave, when struck simultaneously, sound pleasant. Regarding the fifth, Galileo states that "three solitary pulses are separated by intervals of time equal to half the interval which separates each pair of simultaneous beats from the solitary beats of the upper string. Thus the effect of the fifth is to produce a ticking of the ear drum such that its softness is modified with sprightliness, giving at the same moment the impression of a gentle kiss and a bite."

On the other hand, if two notes have a frequency ratio of 167:67, they will sound dissonant. And if the frequency ratio cannot even be expressed as a ratio of integers (it is irrational, like the square root of 2), then the note is most dissonant. As Galileo puts it "…the ear is pained by an irregular sequence of air waves which strike the tympanum without any fixed order."

If you'd like to pursue this topic a bit more, two of my students have sent me interesting resources that you may find interesting—thanks Emily and Elijah! The first is a youtube video explaining the structure and function of the ear and how it acts as an auditory transducer. The second is a short youtube video from minute physics on why it is impossible to tune a piano. You might also check out the excellent book by the famous 19th century doctor and scientists Herman von Helmholtz which is entitled On the Sensations of Tone. Better yet, sit in on a music theory class by Professor William Braun or Jeremy Zima at Wisconsin Lutheran College.